Proceedings of IMECE2009 

2009 ASME International Mechanical Engineering Congress and Exposition 

November 13-19, Lake Buena Vista, Florida, USA 

IMECE2009-10872

PROBABILISTIC APPROACH TO CONDITIONAL PROBABILITY OF RELEASE OF 

HAZARDOUS MATERIALS FROM RAILROAD TANK CARS DURING ACCIDENTS 

D. Y. Jeong 

Volpe National Transportation Systems Center 

US Department of Transportation 

Cambridge, Massachusetts, USA 

ABSTRACT 

This paper describes a probabilistic approach to estimate 

the conditional probability of release of hazardous materials 

from railroad tank cars during train accidents. Monte Carlo 

methods are used in developing a probabilistic model to 

simulate head impacts. The model is based on the physics of 

impact in conjunction with assumptions regarding the 

probability distribution functions of the various factors that 

affect the loss of lading. These factors include impact 

velocity, indenter size, tank material, tank diameter, effective 

collision mass, and tank thickness. Moreover, each factor is 

treated as a random variable characterized by its assumed 

distribution function, mean value, and standard deviation (or 

variance). Reverse engineering is performed to back-calculate 

the mean values and standard deviations of these random 

variables that reproduce trends observed in available accident 

data. The calibrated model is then used to conduct a 

probabilistic sensitivity analysis to examine the relative effect 

of these factors on the conditional probability of release. 

Results from the probabilistic sensitivity analysis indicate that 

the most significant factors that affect conditional probability 

of release are impact velocity, effective collision mass, and 

indenter size. 

INTRODUCTION 

Conditional probability is the probability that an event 

will occur given that another event has already occurred. In 

the present context, conditional probability of release is the 

probability that release of hazardous material from a railroad 

tank car will occur given that an accident (e.g., a train 

derailment or a car-to-car collision) has already occurred. 

Estimates for conditional probability of release have been 

calculated by applying logistic regression analysis [1] to 

accident data contained in the Tank Car Accident Damage 

Database, which is maintained by the Railway Supply Institute 

and the Association of American Railroads (AAR). These 

data have been collected since the 1960s.1

 Here logistic refers 

to the functional form of the mathematical equations used to 

characterize the influence of different factors on the 

probability of lading loss during an accident. Regression 

means that the mathematical equations were curve-fit to 

represent the data. Lading loss is assumed to occur from four 

specific sources: (1) releases from head impacts, (2) releases 

from shell impacts, (3) releases from top fittings, and (4) 

releases from bottom fittings. The regression assumes that the 

most significant factors contributing to the probability of 

lading loss from each source are: (1) whether a head shield is 

present and if so what type, (2) thickness of the tank head, (3) 

whether the tank is jacketed or not, (4) whether double-shelf 

couplers are present, (5) thickness of the tank shell, (6) 

whether the car is pressurized or not, and (7) if the accident 

occurred in a yard or on mainline track. The results of the 

regression analysis and the trends for conditional probability 

of release based on the analysis are described in detail in 

reference [1]. 

A probabilistic approach to examine accidental releases of 

hazardous materials from tank cars is reasonable since no two 

accidents are identical. For example, factors such as load 

severity and material properties tend to vary from one accident 

to the next. Moreover, probabilistic analysis can account for 

randomness and uncertainty in these factors. 

A probabilistic model to predict the percentage of 

accidents that result in brittle fracture and lading loss was 

developed in the 1990s [2, 3]. The model was based on 

probabilistic fracture mechanics analysis in which a 

probability distribution was assumed for cracks or defects that 

may exist in the tank structure (presumably near welds). 

Failure by brittle fracture was assumed when the stress 

intensity factor for a given combination of accident loading 

and defect size exceeded the fracture toughness of the tank car 

 

1

 The database contains over 40,000 records of tank cars damaged in accidents 

[1]. 

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1 

 
steel. Subsequently the model was used to assess the potential 

safety benefit from constructing tank cars with steels 

exhibiting higher strength and toughness properties [4]. 

A probabilistic model is described in this paper that 

estimates conditional probability of release from head impacts 

without the assumption of pre-existing defects or cracks in the 

tank structure. Failure is defined in the model as puncture of 

the commodity tank. Moreover, accident loading and 

puncture resistance of the commodity tank are expressed in 

terms of energy quantities. That is, puncture is assumed to 

occur when the kinetic energy associated with a given impact 

scenario exceeds the energy to puncture (i.e., resistance) for a 

given tank car configuration or design. These energy 

quantities consist of different factors that are treated as 

random variables in the probabilistic model. Mean values and 

standard deviations of the random variables are backcalculated by matching the simulation results to those from the 

logistic regression of the accident data. After the probabilistic 

model is calibrated in this manner, a probabilistic sensitivity 

analysis is performed to examine the relative effect of the 

various factors on conditional probability of release. 

PROBABILISTIC MODEL 

Figure 1 illustrates the fundamental concept in applying 

the probabilistic approach to estimate probability of failure 

[5]. The figures shows two variables; one relating to the 

demand on the system, S (e.g., load on the structure), and the 

other to the resistance of the structure, R. Both S and R are 

considered as random variables that are characterized by their 

mean values S and R, standard deviations S and R, and 

probability density functions fS and fR. Moreover, the overlap 

in the two probability density functions defines the probability 

of failure. If the probability functions are known and can be 

expressed mathematically, the probability of failure can be 

calculated from the following integral 

() () f SR p f sF s



  ds (1) 

where FR is the cumulative probability function representing 

the resistance of the structure. In general, however, exact 

mathematical expressions for the probability density functions 

are unknown. In this paper, Monte Carlo methods are used to 

evaluate the integral in equation (1) to estimate the probability 

of failure. 

Figure 1. Schematic of Probabilistic Approach 

In the probabilistic model described in this paper, the 

demand on the tank car during an accident is defined as the 

kinetic energy associated with the impact while the resistance 

to puncture is defined in terms of energy to puncture. Kinetic 

energy, EK consists of two parts: (1) effective collision mass, 

me and (2) impact velocity, vi,, and is expressed 

mathematically as 

1 2

2 K e i E m  v

c

 (2) 

Puncture resistance in terms of energy to cause puncture 

is assumed to comprise four parts: (1) indenter size in terms 

of the perimeter of the contacting surface P, (2) tank material 

in terms of the ultimate tensile strength of the tank car steel 

 U, (3) tank head thickness h, and (4) tank diameter D. The 

functional form of energy to puncture is assumed as a powerlaw relation: 

a b

P U E CP h D   (3) 

where C, a, b, and c are constants to be determined. For 

dimensional consistency, 

abc    3 (4) 

The functional form of equation (3) for puncture resistance is 

similar to that assumed in semi-empirical analysis of puncture 

during excavation of pipelines [6]. 

Moreover, the factors listed in equations (2) and (3) are 

treated in the model as random variables with assumed 

probability distributions as listed in Table 1. Appendix A lists 

the mathematical expressions for the different probability 

distributions assumed in the model. 

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2 

 
Table 1. Random Variables and Assumed Distributions 

Random Variable Assumed Distribution 

Effective collision mass Weibull 

Impact velocity Weibull 

Indenter size Normal 

Tank material Log Normal 

Tank thickness2 Triangular or Uniform 

Tank diameter Uniform 

Figure 2 shows a schematic of the probabilistic model. 

Monte Carlo methods are used to estimate the conditional 

probability of release. That is, the distribution of accident 

loading is calculated using equation (2) while the distribution 

for the puncture resistance of the tank structure is calculated 

using equation (3). 

Figure 2. Schematic of Simulation Model 

MODEL APPLICATION 

The probabilistic model is applied to produce two 

different sets of results. In the first, tank thickness is treated 

as a deterministic variable with values ranging between 7/16 

and 1 inch. In these cases, the model was exercised to 

simulate 100,000 accidents per thickness value. Mean values 

and standard deviations of the various random variables were 

varied iteratively to reproduce trends observed in the available 

accident data for conditional probability of release as a 

function of thickness. In the second set of results, tank 

thickness is treated as a random variable with an assumed 

probability distribution, and the calibrated model is then used 

to conduct a probabilistic sensitivity analysis to examine the 

relative effect of each factor (or random variable) on the 

conditional probability of release. 

 

2

 Tank thickness is a deterministic variable in the model calibration, and is 

treated as a random variable in the probabilistic sensitivity analysis. 

Table 2. Mean and Standard Deviation for Random 

Variables 

Random Variable Mean,  SD, 

Effective mass, lb-s2/ft 7000 2800 

Impact velocity, mph 30 10 

Indenter perimeter, inches 60 9 

Ultimate tensile strength, ksi 88.8 6.7 

Tank diameter, inches‡ 105 75 

Calibration with Accident Data

Figure 3 shows the trend of conditional probability of 

release as a function of tank head thickness for head impacts 

involving tanks without head shields and without jackets. The 

open symbols represent results from the logistic regression 

analysis of accident data [1]. The dashed lines represent the 

95 percent confidence band on the mean value determined by 

the regression analysis. The solid symbols represent results 

from the Monte Carlo model based on the assumed probability 

distributions listed in Table 1. Table 2 lists the mean values 

and standard deviations for the random variables assumed in 

the model to produce these results. The values for the mean 

and standard deviation for the ultimate tensile strength are 

based on tensile strength measurements conducted on TC128B tank car steel by Southwest Research Institute [7]. The 

other values listed in the table are calculated from applying 

inverse or reverse engineering to match the regression results. 

In addition, the constants in equation (3); namely, C, a, b, and 

c; are calculated from reverse engineering to be equal to 74, 

1.5, 0.5, and 1.0 respectively. The relationships between the 

mean values and standard deviations of the random variables 

and the parameters that characterize the different probability 

functions are given in Appendix A. The results from the 

Monte Carlo simulation show a slightly stronger effect of 

thickness on conditional probability of release than the 

regression on the accident data. Moreover, the simulation 

results are within the 95 percent confidence band for the 

available accident data over the range of thicknesses. 

Figure 4 compares the model results with the regression 

results on the accident data for head losses from jacketed tank 

cars without head shields. The jacket thickness of 0.119 inch 

is added to the tank head thickness to produce the model 

results. In this case, the model overestimates the conditional 

probability of release compared to the accident data for almost 

the entire range of thicknesses. The model results are 

approximately equal to the upper-bound confidence bound at 

thicknesses slightly less than 1 inch, but generally 

overestimate the accident data trend. Similarly, Figure 5

compares results from the model with accident data for tank 

cars with head shields and without jackets. As in the jacketed 

 

‡

 Mean and standard deviation are based on uniform probability density 

function assuming a minimum tank diameter of 90 inches and a maximum of 

120 inches. 

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3 

 
case, the head shield thickness of 0.5 inch is added to the head 

thickness to produce the model results. In this case, the 

simple addition of the shield and tank head thicknesses 

produces results that are within the 95 percent confidence 

bands on observed in available data. 

Probabilistic Sensitivity Analysis

Probabilistic sensitivity analysis is performed using the 

calibrated model for tank heads without head shields and 

without jackets to estimate the relative effect of the various 

factors assumed to affect conditional probability of release. 

Three different measures of probabilistic sensitivity are 

calculated in the present study: (1) probabilistic importance 

factor, (2) probabilistic sensitivity with respect to changes in 

the mean value , and (3) probabilistic sensitivity with respect 

to changes in the standard deviation . The interpretation of 

these sensitivity measures and the specific details of the 

probabilistic sensitivity analysis are described in Appendix B. 

In the probabilistic sensitivity analysis, tank thickness is 

treated as a random variable. Moreover, two distribution 

functions are assumed for tank thickness in the sensitivity 

analysis, triangular and uniform. Table 3 lists the mean values 

and standard deviations for the different distribution functions, 

which are based on assuming a minimum thickness of 7/16 

inch, a maximum of 1 inch, and a mode of 25/32 inch. 

Moreover, these values are calculated using equations listed in 

Appendix A for these assumed distribution functions. 

Table 3. Mean Values and Standard Deviations for Tank 

Thickness and Different Distribution Functions 

Assumed Distribution Mean,  SD, 

Triangular 0.771 0.127 

Uniform 0.719 0.026 

Figure 6 shows the probabilistic importance factors for 

the six random variables assumed in the Monte Carlo model to 

estimate the conditional probability of release. Results are 

shown for the two different assumptions regarding the 

probability distribution for thickness. Similarly, Figure 7

shows the probabilistic sensitivities with respect to changes in 

the mean of the random variables. Figure 8 shows the 

probabilistic sensitivities with respect to changes in standard 

deviation. The length of the bars in these figures indicates the 

relative effect of the various random variables on the 

conditional probability of release. For example, each measure 

of probabilistic sensitivity indicates that impact velocity has 

the most significant effect on the estimate of conditional 

probability of release. 

Head Thickness (inch)

0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

Conditional Probability of Lading Loss

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

Monte Carlo Simulation

Regression on Accident Data (RA 05-02)

95% Confidence Band

No Head Shield and No Jacket

Figure 3. Conditional Probability of Release for Head 

Losses (No Head Shield and No Jacket) 

Head Thickness (inch)

0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

Conditional Probability of Lading Loss

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

Monte Carlo Simulation

Regression on Accident Data (RA 05-02)

95% Confidence Band

No Head Shield, Jacketed

Figure 4. Conditional Probability of Release for Head 

Losses (No Head Shield, Jacketed) 

Head Thickness (inch)

0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

Conditional Probability of Lading Loss

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

Monte Carlo Simulation

Regression on Accident Data (RA 05-02)

95% Confidence Band

Head Shield, No Jacket

Figure 5. Conditional Probability of Release for Head 

Losses (Head Shield, No Jacket) 

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Random Variable

Effective mass

Impact velocity

Indenter size

Material

Thickness

Diameter

Probabilistic Importance Factor

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Triangular

Uniform

Figure 6. Probabilistic Importance Factors 

Random Variable

Effective mass

Impact velocity

Indenter size

Material

Thickness

Diameter

Probabilistic Sensitivity Factor

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

Triangular

Uniform

Figure 7. Probabilistic Sensitivity with Respect to Changes 

in Mean 

Random Variable

Effective mass

Impact velocity

Indenter size

Material

Thickness

Diameter

Probabilistic Sensitivity Factor

0.0

0.5

1.0

1.5

2.0

Triangular

Uniform

Figure 8. Probabilistic Sensitivity with Respect to Changes 

in Standard Deviation 

Moreover, the relative ranking of the random variables is 

consistent for each measure of probabilistic sensitivity. 

However, the ranking order between tank thickness and tank 

diameter varies depending on the assumed distribution 

function for thickness (Table 4). Regardless of the assumed 

distribution for thickness, the probabilistic sensitivity analysis 

indicates that the three most significant factors affecting 

conditional probability of release are impact velocity, effective 

collision mass, and indenter size. 

Table 4. Relative Ranking of Factors Affecting Conditional 

Probability of Release 

Assumed Distribution for Tank Thickness 

Triangular Uniform 

(1) Impact velocity Impact velocity 

(2) Effective mass Effective mass 

(3) Indenter size Indenter size 

(4) Tank diameter Tank thickness 

(5) Tank thickness Tank diameter 

(6) Material Material 

DISCUSSION 

In this paper, results from the probabilistic model are 

calibrated to produce trends that are consistent with those 

observed in accident data as tank head thickness is varied. 

That is, conditional probability of release from head impacts 

tends to decrease as head thickness increases. The calibration 

procedure entails inverse or reverse engineering to backcalculate mean values and standard deviations for the random 

variables assumed in the model as well as constants for 

puncture energy. Model results are within the 95 percent 

confidence bounds of the logistic regression results from the 

available accident data for tank heads without head shields 

and without jackets (Figure 3). The effect of jackets and head 

shields is incorporated into the model by adding jacket/shield 

thickness to head thickness. The discrepancy between model 

results and accident data for jacketed tank heads without head 

shields (Figure 4) suggests that other factors may contribute to 

the impact mechanics which have not been addressed in the 

probabilistic model. This hypothesis, however, is not 

supported in the case of unjacketed tank heads with head 

shields where the results from the model without accounting 

for other factors are within excellent agreement with the 

accident data (Figure 5). 

Probabilistic sensitivity analysis indicates that the most 

significant factors affecting conditional probability of release 

are impact velocity, effective collision mass, and indenter size. 

The relative ranking of these factors is based upon the various 

assumptions made in developing the model, but appears 

reasonable. For example, the strong effect of speed is 

consistent with results from a previous study on the 

quantitative estimation of risk associated with the 

transportation of hazardous materials (hazmat) by rail [8]. In 

the previous study, speed is shown to have a three-fold effect 

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5 

 
on hazmat accidents. As speed increases: (1) the number of 

derailing cars increases, thus increasing the probability that 

hazmat cars present in the train will derail, (2) the probability 

that a derailed hazmat car will release its contents increases, 

and (3) the amount of hazmat from a releasing car increases. 

The strong effect of indenter size is consistent with results 

from deterministic sensitivity studies of puncture resistance to 

tank car shell impacts based on finite element analysis [9]. In 

addition, deterministic sensitivity analysis of pipeline puncture 

during excavations indicates that puncture is strongly 

dependent on tooth size (i.e., indenter size), pipe wall 

thickness, and material grade [10, 11]. 

Two of the three factors having the most significant effect 

of conditional probability of release; namely, effective 

collision mass and indenter size; are often unknown in actual 

accidents. For example, in composite summaries of major 

railroad accidents involving the release of hazardous materials 

between 1969 and 1978 [12], the probable impacting object 

was reported in 22 of 75 accidents. In these 22 accidents, 13 

were couplers, five were drawbars, two were broken rails, one 

was a wheel, and one was an end sill of a car. 

The relatively weak effect of thickness, compared to other 

factors, is examined by assuming different probability 

functions, specifically triangular and uniform. The ranking 

order between tank thickness and tank diameter is altered 

depending on the assumed distribution, but the order of the 

three most significant factors affecting conditional probability 

of release is unchanged. 

The use of conditional probability of release as a 

performance metric stems from its direct applicability to 

perform risk assessments and cost/benefit analysis. However, 

its usefulness may become problematic if and when enhanced 

designs that apply technologies beyond those used in currently 

accepted practice (e.g., impact energy absorbing structures) 

become practical alternatives to conventional designs. 

Estimates for the conditional probability of release of 

hazardous materials from tank car with alternative designs 

must be based on extrapolations and simplifying assumptions 

initially due to the lack of a sufficient database of accident and 

failure experience. 

Conditional probability of release considers the 

probability of lading loss but does not take into account the 

amount of lost commodity during the accidental release of 

hazardous material. Moreover, no attempt has been made in 

this paper to examine the quantity of lading loss in accidents. 

CONCLUDING REMARKS 

This paper describes a probabilistic approach to estimate 

conditional probability of release of hazardous materials from 

tank car head impacts during accidents. Monte Carlo methods 

are used in the model to reproduce trends that mimic those 

observed in available accident data compiled over a forty-year 

period. A probabilistic sensitivity analysis is performed with 

the calibrated model to examine the relative effect of the 

various factors assumed in the model on conditional 

probability of release. Sensitivity is calculated on the basis of 

three different measures. The relative ranking of the most 

significant factors affecting conditional probability of release 

is consistent for each measure of probabilistic sensitivity. 

Moreover, results from the probabilistic sensitivity analysis 

suggest that the three most significant factors that affect 

conditional probability of release are impact velocity, effective 

collision mass, and indenter size. 

ACKNOWLEDGMENTS 

The research described in this paper was sponsored by the 

Federal Railroad Administration, Office of Research and 

Development under the Rail Equipment Safety Research 

Program. Mr. Francisco Gonzalez III of the Equipment and 

Operating Practices Division is the Program Manager for 

research related to railroad tank car safety. 

REFERENCES 

[1] Treichel, T., Hughes, J.P., Barkan, C.P.L., Sims, R.D., 

Phillips, E.A., Saat, M.R., Wen, Y.K., Simpson, D.G., 

“Safety Performance of Tank Cars in Accidents: 

Probability of Lading Loss,” RSI-AAR Railroad Tank 

Car Safety Research and Test Project Report No. RA 05-

02, January 2006. 

[2] Hughes, J.P., Heuer, C.F., Anderson, T.L., “Fracture 

Behavior of Tank Car Steels in Accidents from 1981 

through 1994,” RSI-AAR Railroad Tank Car Safety 

Research and Test Project Report No. RA 03-6-62, 

December 1998. 

[3] Anderson, T.L., “Probabilistic Fracture Mechanics 

Analysis of Lading Loss in Railroad Tank Cars,” 43rd

Mechanical Working and Steel Processing Conference 

Proceedings, Iron and Steel Society, Vol. XXXIX, 

October 2001, pp. 883-894. 

[4] Sims, R.D., Hughes, J.P., “Steels for Tank Cars,” 43rd

Mechanical Working and Steel Processing Conference 

Proceedings, Iron and Steel Society, Vol. XXXIX, 

October 2001, pp. 825-835. 

[5] Haldar, A., Mahadevan, S., Probability, Reliability and 

Statistical Methods in Engineering Design, John Wiley 

& Sons, 2000. 

[6] Palmer, A., Neilson, A., Sivadasan, S., “Impact 

resistance of pipelines and the loss-of-containment limit 

state,” Journal of Pipeline Integrity 2, 2003, pp. 231-

240. 

[7] McKeighan, P.C., “Mechanical Properties of Tank Car 

Steels Retired from the Fleet,” Southwest Research 

Institute Report to Volpe Center, August 2008. 

[8] Nayak, P.R., Rosenfield, D.B., Hagopian, J.H., “Event 

Probabilities and Impact Zones for Hazardous Materials 

Accidents on Railroads,” Final Report No. 

DOT/FRA/ORD-83/20, November 1983. 

[9] Tang, Y.H., Yu, H., Gordon, J.E., Jeong, D.Y., “Analysis 

of Railroad Tank Car Shell Impacts Using Finite Element 

This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States. Approved for public release; distribution is unlimited. 

6 

 
Method,” Proceedings of the 2008 IEEE/ASME Joint 

Rail Conference, JRC2008-63014, April 2008. 

[10] Brooker, D.C., “Numerical modeling of pipeline 

puncture under excavator loading, Part I: development 

and validation of finite element material failure model 

for puncture simulation,” International Journal of 

Pressure Vessels and Piping 80, 2003, pp. 715-725. 

[11] Brooker, D.C., “Numerical modeling of pipeline 

puncture under excavator loading, Part II: parametric 

study,” International Journal of Pressure Vessels and 

Piping 80, 2003, pp. 727-735. 

[12] Glickman, T.S., “Major Railroad Accidents Involving 

Hazardous Materials Release, Composite Summaries, 

1969-1978,” Report No. FRA-RRS-80-04, July 1980. 

[13] Ayyub, B.M., McCuen, R.H., Probability, Statistics & 

Reliability for Engineers, CRC Press, 1997. 

[14] Wu, Y.T., “Computational methods for efficient 

structural reliability and reliability sensitivity analysis,” 

AIAA Journal 32, August 1994, pp. 1717-1723. 

[15] Mohanty, S., Wu, Y.T., “CDF sensitivity analysis 

technique for ranking influential parameters in the 

performance assessment of the proposed high-level 

waste repository at Yucca Mountain, Nevada, USA,” 

Reliability Engineering and System Safety 73, 2001, pp. 

167-176. 

APPENDIX A – PROBABILITY FUNCTIONS 

This appendix lists the probability density functions 

(PDF) and cumulative distribution functions (CDF) for the 

different distributions considered in the probabilistic model 

described in this paper. 

Normal Distribution

The PDF for a normal or Gaussian distribution is 

 2

2

1

( ) exp

2 2

x

f x 

  

   



  (A-1) 

where  is the mean value and  is the standard deviation. 

The corresponding CDF is 

    2

2

1

( ) exp

2 2

x x x

F x dx  

    

            

   



 (A-2) 

where  is the standard error function which is defined as 

1 1 2

( ) exp

2 2

z

z  d

 

         (A-3) 

Log Normal Distribution

The PDF for a log normal distribution is 

 2

2

1 ln( ) ( ) exp

2 2

x

f x

x



  

       

 (A-4) 

where  and  are parameters that are related to the mean 

and standard deviation  through the following equations: 

. 

1 2

exp

2

       

  (A-5) 

   2 2     exp 2 exp 1       (A-6) 

The corresponding CDF for a log normal distribution can be 

expressed in terms of the standard normal integral as 

ln( ) ( )x

F x





        

 (A-7) 

Weibull Distribution

The three-parameter Weibull probability distribution is 

defined as 

 

1

( ) exp

x x

f x

 

 

  



     

           (A-8) 

where  is called the shape parameter,  is called the scale 

parameter, and  is called the location parameter. For a twoparameter Weibull distribution, the location parameter  is 

equal to zero, and the Weibull shape and scale parameters are 

related to the mean value  and the standard deviation 

through the following relations: 

1

  1



     

  (A-9) 

2

2 1

  1 1

 

                     (A-10) 

where  is the complete gamma function which is defined as 

  1

0

x

x e d  



     (A-11) 

The CDF for the three-parameter Weibull distribution is 

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7 

 
( ) 1 exp

x

F x





 

   

            (A-12) 

Triangular Distribution

The PDF for a triangular distribution is 

 

  

 

  

min

min mode

mode min max min

max

mode max

max mode max min

2

if

( )

2

if

x x

x xx

x xxx

f x

x x

x x x

xx xx

          













 (A-13) 

where x must lie between xmin and xmax, and xmode is the most 

likely value. For the triangular probability distribution, the 

mean and standard deviation are related to the minimum, 

maximum, and most likely values through the following 

equations: 

. 

 min mode max

1

3

   xx x  (A-14) 

  22 2

min mode max min mode min max mode max

1

18

      x x x xx xx x x 

 (A-15) 

The corresponding CDF for a triangular distribution is 

 

  

 

  

min

2

min

min mode

mode min max min

2

max

mode max

max mode max min

max

0 if

if

( )

1 if

1 if

x x

x x

x x x

x xxx

F x

x x

x x x

xx xx

x x



     

    























(A-16) 

Uniform Distribution

The PDF for a uniform distribution is 

  min max

max min

1

if

( ) 16

0 otherwise

x x x

f x x x

   











 (A-17) 

The mean value and standard deviation are calculated from the 

minimum and maximum values: 

 min max

1

2

   x x  (A-18) 

 1 2

12 max min    x x  (A-19) 

The corresponding CDF is 

min

min

min max

max min

max

0 if

( ) if

1 if

x x

x x

F x x

x x

x x



  

















 x  x (A-20) 

APPENDIX B – PROBABILISTIC SENSITIVITY 

This appendix summarizes the concepts and describes the 

mathematical calculations that were carried out to perform the 

probabilistic sensitivity analysis in this paper. 

In probabilistic structural analysis, the measure of risk is 

commonly expressed in terms of the probability of failure. 

The probability of failure of a structure is determined from a 

specific performance criterion that depends on the relevant 

loading and structural resistance (i.e. strength) parameters that 

are treated as random variables. A response function is 

generally defined as a function of a vector of random variables 

1 2 () ( , , ) n Z X ZX X X   (A-21) 

where the Xi’s are the random variables and X represents the 

vector of random variables. The corresponding performance 

function defines the failure surface or limit-state that 

characterizes the assumed failure criterion 

0 g ZX Z  () 0   . (A-22) 

The performance function or limit-state function is written 

such that values of g less than zero represent failure. 

Therefore, the probability of failure is related to the 

performance function by 

  ( )0

0 X

g X

p Pg f x dx      ( ) (A-23) 

where fX is the joint probability density function. 

In this paper, the multiple integral in equation (A-23) is 

evaluated through Monte Carlo procedures. Moreover, the 

performance function assumed is defined as 

p K g   E E (A-24) 

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8 

 
where Ep is the puncture resistance in terms of energy and EK

is the applied accident loading in terms of kinetic energy. The 

factors affecting energy to puncture or the resistance against 

puncture in terms of energy and their functional relation are 

expressed through the following equation 

a b

P U E CP h D   c

 (A-25) 

where P is the perimeter of the indenter,  U is the ultimate 

tensile strength of the tank car steel, h is the tank thickness, 

and D is the tank diameter. In addition, C, a, b, and c are 

constants. The functional form of this equation for puncture 

resistance is similar to that used in semi-empirical analysis of 

puncture during excavation of pipelines [6]. 

The factors affecting kinetic energy and their functional 

relation are given by 

1 2

2 K e i E m  v (A-26) 

where me is the effective collision mass and vi is the impact 

velocity. 

In general, the following mathematical problem must be 

solved: 

Minimize  2

1

n

i

i

 x 

   such that g x() 0   (A-27) 

where x’ refers to the vector of reduced random variables, 

which is defined below. Moreover, equation (A-27) is usually 

solved using nonlinear optimization methods. Alternatively, 

the solution can be achieved iteratively by solving the 

following set of simultaneous equations in an iterative 

procedure: 

*

2

1 * *

i

n

i i i

g

X X 



  

   

     

g (A-28) 

* *

i i x    (A-29) 

 ** *

1 2 , n gx x x      0 . (A-30) 

The subscript or superscript of * means that quantity or 

expression is evaluated at the design point. Mathematically, 

equation (A-28) represents the directional cosines of the 

tangent hyperplane in the reduced coordinates. The 

directional cosines are indicators of the sensitivity of the 

performance function to variations in the random variables. 

The following computational steps can be applied to solve 

equations (A-28) to (A-30), which is described by Ayyub and 

McCuen [13] as an advanced first-order second-moment 

(AFOSM) method: 

1. Assume an initial value for the design point. In the present 

analysis, the mean values of the random variables were chosen 

as the initial guess. The design point in the reduced 

coordinates is calculated using 

' i

i

i X

i i

X

x u 



 x   . (A-31) 

2. Evaluate the directional cosines at the failure point. The 

partial derivatives to calculate the directional cosines are 

given by 

* **

i

i

X

i ii i

gg g X

X XX X *

         

                     (A-32) 

3. Solve the following equation for the root : 

      11 2 2

** *

1 2

, 0 n n X X X X Xn Xg              

i

 (A-33) 

4. Using the value of  from step 3, evaluate a new design 

point using the following equation: 

* *

i i X iX x      (A-34) 

5. Repeat steps 1 to 4 until convergence of  is obtained. The 

value of  is referred to as the reliability index. 

Sensitivity measures can be used to quantify the influence of 

each random variable on the probability of failure. One 

measure of sensitivity is called the probabilistic importance 

factor, which is equivalent to the normalized gradient vector 

of the performance function in the space of the standard 

normal variables 

g

g







 (A-35) 

This equation is equivalent to the vector for the direction 

cosines of the random variables; i.e., equation (A-28). The 

direction cosines are also related to the reliability index: 

*

*

i

i

i X

u

u







   

 (A-36) 

The vector for the probabilistic importance factors also has the 

property that 

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9 

 
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10 

 

22 2

1 2 1   n     (A-37) [ ( )] ( )

i i i

p

i

   

   

  

   



       (A-41) 

where n is the number of random variables. Moreover, the 

probabilistic importance factors, i, are used as one of the 

sensitivity measures in this document to determine the relative 

ranking of the random variables in the Monte Carlo model for 

tank cars in accidents. 

where  is the standard normal probability density function. 

Then applying the chain rule of differentiation, 

() () i

i

i ii

p u

u

     i

i

u

  

           

 (A-42) 

Two other probabilistic sensitivity measures have been 

developed by Wu [14, 15] that calculate the change in the 

probability of failure, p, with respect to changes in the mean 

value and the standard deviation of the random variables: 

where 

/

/ i

i i

p p S

 

 



 (A-38) 

i

i

i ii

u

u

 

 i

i

u

  

        

 (A-43) 

/

/ i

i i

p p S

 

 



 (A-39) 

Similarly, the derivative of equation (A-40) with respect to the 

standard deviation  is 

[ ( )] ( )

i i i

p

i

   

   

  

   



     

 (A-44) 

where i and i are the mean and standard deviation, 

respectively, of the random variables. The use of the standard 

deviation as a scale factor in these sensitivity measures 

implies that the allowable range of the mean value is limited 

to a local region characterized by the uncertainty of the 

random variable. Moreover, these sensitivity measures are 

dimensionless and can take positive, negative, or zero values. 

When S is zero or relatively small, it implies that the 

corresponding random variable can be varied over a wide 

range without significantly changing the probability of failure. 

This, in turn, implies that S will be relatively small. On the 

other hand, when S is relatively large, the corresponding S

will also tend to be large, which suggests that S and S are 

strongly related and both can be used to identify key 

contributing random variables. For a single limit state, i and 

Si may be approximately proportional to i if ui

* are 

relatively large. 

Applying the chain rule, 

() () i

i

i ii

p u

u

     i

i

u

  

           

 (A-45) 

where 

i

i

i ii

u

u

 

 i

i

u

  

        

 (A-46) 

Therefore, combining equations gives 

( )

( ) i

i

i

i i

p u S

p 

   i   i  

    

  

      

 (A-47) In order to compute S and S, equations (A-38) and 

(A-39) are expressed as follows. The probability of failure is 

related to the reliability index through the following equation 

p     1 () (   ) (A-40) 

( )

( ) i

i

i

i i

p u S

p 

   i  i 

 

    

  

      

 (A-48) 

where  is the standard normal cumulative probability 

function, which was defined in equation (A-3). Equation 

(A-40) is differentiated with respect to the mean value 

Classical perturbation methods are applied to evaluate the 

partial derivatives of u with respect to the mean and standard 

deviation.